Classification of Groups of Order up to 15

Theorem

Up to isomorphism, every group of order $|G| \le 15$ is one of the below:

Order	Abelian	Non-Abelian
1	$\Z_1$
2	$\Z_2$
3	$\Z_3$
4	$\Z_4, \Z_2 \oplus \Z_2$
5	$\Z_5$
6	$\Z_6$	$D_3$
7	$\Z_7$
8	$\Z_8, \Z_4 \oplus \Z_2, \Z_2 \oplus \Z_2 \oplus \Z_2$	$D_4, Q_4$
9	$\Z_9, \Z_3 \oplus \Z_3$
10	$\Z_{10}$	$D_5$
11	$\Z_{11}$
12	$\Z_{12}, \Z_6 \oplus \Z_2$	$D_6, A_4, Q_6$
13	$\Z_{13}$
14	$\Z_{14}$	$D_7$
15	$\Z_{15}$

where:

• $D_n$ is the dihedral group of order $2n$;
• $A_n$ is the alternating group on $n$ points;
• $Q_n$ is the dicyclic group on $2n$.

Proof

The Abelian cases are the direct result of the Fundamental Theorem of Finite Abelian Groups.

The non-Abelian cases follow from seven separate theorems:

$(1): \quad$ Trivial Group - determines theorem for order $1$

$(2): \quad$ Group of Prime Order Cyclic - determines theorem for orders $2$, $3$, $5$, $7$, $11$, and $13$

$(3): \quad$ Group of Order Prime Squared is Abelian - determines theorem for orders $4$ and $9$

$(4): \quad$ Cyclic Groups of Order pq - determines theorem for order $15$

$(5): \quad$ Groups of Order Twice a Prime - determines theorem for orders $6$, $10$, $14$

$(6): \quad$ Groups of Order 8 - determines theorem for order $8$

$(7): \quad$ Groups of Order 12 - determines theorem for order $12$